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Difference between revisions of "User:Boris Tsirelson/sandbox1"

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The term "universally measurable" may be applied to
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* a [[measurable space]];
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* a subset of a measurable space;
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* a [[metric space]].
  
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Let $(X,\A)$ be a measurable space. A subset $A\subset X$ is called universally measurable, if it is $\mu$-measurable for every finite measure $\mu$ on $(X,\A)$. In other words: $\mu_*(A)=\mu^*(A)$ where $\mu_*,\mu^*$ are the inner and outer measures for $\mu$, that is,
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: $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
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\mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$
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====References====
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{|
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|valign="top"|{{Ref|T}}||  Terence Tao, "An introduction to measure theory", AMS (2011).    {{MR|2827917}}   {{ZBL|05952932}}
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|-
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|valign="top"|{{Ref|P}}||  David Pollard, "A user's guide to measure theoretic probability",  Cambridge (2002).    {{MR|1873379}}    {{ZBL|0992.60001}}
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|-
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|valign="top"|{{Ref|K}}|| Alexander  S.  Kechris, "Classical  descriptive set theory", Springer-Verlag  (1995).    {{MR|1321597}}   {{ZBL|0819.04002}}
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|-
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|valign="top"|{{Ref|BK}}||  Howard Becker and Alexander S. Kechris, "The descriptive set theory of  Polish group actions", Cambridge (1996).   {{MR|1425877}}     {{ZBL|0949.54052}}
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|-
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|valign="top"|{{Ref|D}}||  Richard M. Dudley, "Real analysis and probability",  Wadsworth&Brooks/Cole (1989).   {{MR|0982264}}    {{ZBL|0686.60001}}
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|-
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|valign="top"|{{Ref|M}}||  George  W.  Mackey,  "Borel structure in groups and their duals",  ''Trans.  Amer.  Math. Soc.''  '''85''' (1957), 134–165.   {{MR|0089999}}    {{ZBL|0082.11201}}
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|-
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|valign="top"|{{Ref|H}}||  Paul R. Halmos, "Measure theory", v. Nostrand (1950).    {{MR|0033869}}   {{ZBL|0040.16802}}
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|-
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|valign="top"|{{Ref|R}}||  Walter Rudin, "Principles of mathematical analysis", McGraw-Hill  (1953).   {{MR|0055409}}   {{ZBL|0052.05301}}
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|}

Revision as of 19:35, 16 February 2012

$\newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ The term "universally measurable" may be applied to

Let $(X,\A)$ be a measurable space. A subset $A\subset X$ is called universally measurable, if it is $\mu$-measurable for every finite measure $\mu$ on $(X,\A)$. In other words: $\mu_*(A)=\mu^*(A)$ where $\mu_*,\mu^*$ are the inner and outer measures for $\mu$, that is,

$ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad \mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$

References

[T] Terence Tao, "An introduction to measure theory", AMS (2011).   MR2827917   Zbl 05952932
[P] David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002).   MR1873379   Zbl 0992.60001
[K] Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[BK] Howard Becker and Alexander S. Kechris, "The descriptive set theory of Polish group actions", Cambridge (1996).   MR1425877   Zbl 0949.54052
[D] Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001
[M] George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[H] Paul R. Halmos, "Measure theory", v. Nostrand (1950).   MR0033869   Zbl 0040.16802
[R] Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953).   MR0055409   Zbl 0052.05301
How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21102